Reinforcement Learning for Multiple Agents

I already have a functioning $Q(\lambda)$ implementation for a single agent working on a dynamic pricing problem with the goal of maximizing revenue. The problem that I'm working with, however, involves several different products that are replacements for each other, so dynamically pricing them all with independent learners seems incorrect, because the price of one influences the reward of the other. The goal would be to dynamically price them all so as to maximize the sum of each individual revenue.

I've been doing some research to try to find something that applies reinforcement learning in this way, but many multi-agent implementations I have found focus more on competitive games than cooperative, or they assume incomplete knowledge of other agents (I would have complete knowledge of each agent in this scenario). Are there any well-researched/documented applications of cooperative learning in this way?

Edit: After doing a little more research, I've found some papers that propose that problems with heterogeneous agents that can communicate effectively can essentially be reduced to a single-agent Q-learning problem that controls the other agents. However, I can't seem to find any steps for implementation.

• Here's a paper that might be useful: "Complex dynamics in learning complicated games" pnas.org/content/110/4/1232.full. They implement a type of reinforcement learning called "Experience Weighted Attraction" that they say works for N-player games, though they only implement 2-player games. Their space of games include cooperative payoffs, but they don't model shared information among players. Their results show what conditions lead to a single equilibrium, many equilibria, and none (i.e. chaotic attractor for player dynamics). – MrMeritology Jul 12 '15 at 0:15
• Could you possibly provide a little bit of mathematical formulation of your problem? To me, it seems that this could be related to the oligopoly-cartel model thismatter.com/economics/oligopoly-pricing-models.htm . – Karel Macek Jul 28 '15 at 6:40
• Any updates on this ? – Ic3fr0g Feb 8 at 8:06